{
  "abstract": "We study how amphichirality of a knot constrains its braid index and, in the smallest nontrivial case, the combinatorics of its braid words. Using the Dynnikov--Prasolov resolution of the Jones conjecture, we show the braid index of an amphichiral knot is odd. This answers, in the negative, a question of Stoimenow on amphichiral knots of even braid index. We then classify prime amphichiral knots of braid index~$3$. Every amphichiral knot of braid index~$3$ is alternating and admits a minimal $3$-braid representative in a standard form encoded by a word~$c$. Building on the Birman--Menasco classification of closed $3$-braids and the Murasugi normal form, we describe a dihedral action on~$c$ under which the mirror and mirror-reverse operations are realized by a rotation and a reflection. For a standard form whose closure is a prime knot of braid index~$3$, this yields a complete criterion: $\\widehat{β_c}$ is amphichiral if and only if $c$ is a palindrome or has odd period, together with a determination of the precise symmetry type in terms of these properties and the presence of a non-degenerate flype.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12295",
  "author_total": 1,
  "authors": [
    {
      "name": "Hyungseok Jung"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c1swz7636c4vn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12295",
  "fulltext_url": null,
  "id": "c1swz7636c4vn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12295",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7212592710",
  "pdf_url": "https://arxiv.org/pdf/2609.12295",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c1swz7636c4vn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12295",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12295"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case",
  "title_text": "Amphichiral Knots: Odd Braid Index and Symmetry Classification in the Three-Braid Case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/c1swz7636c4vn",
  "venue": "arXiv",
  "versions": []
}